Bound states and the effective potential in field theories

We calculate the leading contribution to the effective potential, V (phi), under the assumptions that there is a deep bound state and the corresponding pole dominates the one-particle irreducible vertices for spin-zero field theories. We find that V has a 3/2-power branch point in the phi plane on the real axis. We show that the 3/2 power is due to the fact that V'=dV/dphi satisfies an algebraic equation quadratic and cubic in V' for the cases considered. In the domain of phi for which V is real, the leading pole contribution is negative, allowing for the possibility of an instability in the normal vacuum